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Properties

Label 2-2217-2217.845-c0-0-0
Degree $2$
Conductor $2217$
Sign $-0.581 + 0.813i$
Analytic cond. $1.10642$
Root an. cond. $1.05186$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.477 − 0.878i)3-s + (−0.771 − 0.636i)4-s + (0.552 − 1.53i)7-s + (−0.543 − 0.839i)9-s + (−0.927 + 0.373i)12-s + (1.60 − 0.795i)13-s + (0.190 + 0.981i)16-s + (1.88 + 0.596i)19-s + (−1.08 − 1.22i)21-s + (−0.543 + 0.839i)25-s + (−0.997 + 0.0765i)27-s + (−1.40 + 0.835i)28-s + (−0.974 + 0.482i)31-s + (−0.114 + 0.993i)36-s + (0.0145 + 0.0752i)37-s + ⋯
L(s)  = 1  + (0.477 − 0.878i)3-s + (−0.771 − 0.636i)4-s + (0.552 − 1.53i)7-s + (−0.543 − 0.839i)9-s + (−0.927 + 0.373i)12-s + (1.60 − 0.795i)13-s + (0.190 + 0.981i)16-s + (1.88 + 0.596i)19-s + (−1.08 − 1.22i)21-s + (−0.543 + 0.839i)25-s + (−0.997 + 0.0765i)27-s + (−1.40 + 0.835i)28-s + (−0.974 + 0.482i)31-s + (−0.114 + 0.993i)36-s + (0.0145 + 0.0752i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (845, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ -0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297958230\)
\(L(\frac12)\) \(\approx\) \(1.297958230\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.477 + 0.878i)T \)
739 \( 1 + (0.927 - 0.373i)T \)
good2 \( 1 + (0.771 + 0.636i)T^{2} \)
5 \( 1 + (0.543 - 0.839i)T^{2} \)
7 \( 1 + (-0.552 + 1.53i)T + (-0.771 - 0.636i)T^{2} \)
11 \( 1 + (0.264 - 0.964i)T^{2} \)
13 \( 1 + (-1.60 + 0.795i)T + (0.606 - 0.795i)T^{2} \)
17 \( 1 + (-0.338 + 0.941i)T^{2} \)
19 \( 1 + (-1.88 - 0.596i)T + (0.817 + 0.575i)T^{2} \)
23 \( 1 + (0.771 + 0.636i)T^{2} \)
29 \( 1 + (0.973 + 0.227i)T^{2} \)
31 \( 1 + (0.974 - 0.482i)T + (0.606 - 0.795i)T^{2} \)
37 \( 1 + (-0.0145 - 0.0752i)T + (-0.927 + 0.373i)T^{2} \)
41 \( 1 + (-0.896 - 0.443i)T^{2} \)
43 \( 1 + (1.97 + 0.151i)T + (0.988 + 0.152i)T^{2} \)
47 \( 1 + (0.859 - 0.511i)T^{2} \)
53 \( 1 + (0.927 - 0.373i)T^{2} \)
59 \( 1 + (-0.988 + 0.152i)T^{2} \)
61 \( 1 + (0.253 - 0.284i)T + (-0.114 - 0.993i)T^{2} \)
67 \( 1 + (-0.838 - 1.29i)T + (-0.409 + 0.912i)T^{2} \)
71 \( 1 + (0.859 + 0.511i)T^{2} \)
73 \( 1 + (-0.444 - 0.992i)T + (-0.665 + 0.746i)T^{2} \)
79 \( 1 + (-0.0258 + 0.675i)T + (-0.997 - 0.0765i)T^{2} \)
83 \( 1 + (0.997 + 0.0765i)T^{2} \)
89 \( 1 + (-0.606 - 0.795i)T^{2} \)
97 \( 1 + (0.627 - 1.74i)T + (-0.771 - 0.636i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.832605094427509750779415337707, −8.106480469914445805109464964408, −7.56777248419243896865047855549, −6.75320304385474003707823311988, −5.76459248861838454499079799905, −5.12476286379618942637986672356, −3.73168207756083227853482984129, −3.49465282034905097242960626281, −1.49346370920448228144141340265, −1.03730958193828819899243857777, 1.91641240815725267038760149107, 3.08401618952324078347682042792, 3.70772656347759838944106478081, 4.72785588486426248271781569209, 5.31300276898910309147708871937, 6.14094687544062631366420132296, 7.51635143594177077164160494102, 8.352946380506356231367231141476, 8.700734012912062357323055176342, 9.356647246823350655777216507309

Graph of the $Z$-function along the critical line